Research seriesL3algorithms
ML-KEM and ML-DSA rest on one family of assumptions — Learning With Errors and its structured variants. The lattice problems, the worst-case-to-average-case reductions, and the concrete parameter methodology, derived from scratch. Grounded in the primary papers.
The post-quantum standards rest on a geometry: finding short or near vectors in a high-dimensional grid is easy with a good basis and hard with a bad one.
Add a little noise to a system of linear equations and recovering the secret becomes as hard as the worst lattice problem. That noisy-equation problem is the engine of the post-quantum standards.
Plain LWE is secure but heavy. Wrapping it in a polynomial ring shrinks keys from quadratic to linear; the module variant then dials structure up and down to hedge the risk that buys.
How do you turn 'believed hard' into a specific dimension and modulus? By costing the best known lattice attack and choosing parameters that push it past a target like 2^143 operations.
The standards are not broken, but their extra algebraic structure is a real attack surface. The honest question is which regimes are dangerous — and why the deployed parameters stay out of them.